From Ebbinghaus, Flum, and Thomas' Mathematical Logic, chapter 2, exercise 1.3:

Let $\alpha : \mathbb{N} \to \mathbb{R}$ be given. For $a, b \in \mathbb{R}$ such that $a < b$ show that there is a point $c$ in the closed interval $I = [a,b]$ such that $c \notin \{\alpha(n) : n \in \mathbb{N}\}$. Conclude from this that $I$, and hence $\mathbb{R}$ also, are uncountable. (Hint: By induction define a sequence $I = I_0 \supset I_1 \supset \dots$ of closed intervals such that $\alpha(n) \notin I_{n+1}$ and use the fact that $\bigcap_{n \in \mathbb{N}} I_n \ne \varnothing$.)

Define $I_0 = [a,b]$. For any $n \geq 0$, if $I_n = [a_n, b_n]$, let $m = \frac{a_n+b_n}{2}$ and define

For any $n \geq 0$, we can see that for any value of $\alpha(n)$, we have $\alpha(n) \notin I_{n+1}$, so $\alpha(n) \notin \bigcap_{k\in\mathbb{N}} I_k$ Therefore, if $c \in \bigcap_{n\in\mathbb{N}} I_n$, then $c \neq \alpha(n)$ for any $n \in \mathbb{N}$.

As $\bigcap_{n\in\mathbb{N}} I_n \neq \varnothing$ there is some $c \in \bigcap_{n\in\mathbb{N}} I_n$, and for this $c$, we have $c \in I_0=[a,b]$ (and $c \in \mathbb{R}$) and $c \neq \alpha(n)$ for any $n \in \mathbb{N}$, so for any given $\alpha$, then $\alpha$ is not surjective onto $[a,b]$ (and $\mathbb{R}$). Therefore, $[a,b]$ (and $\mathbb{R}$) is not countable.